Mod10 Control Chart

7/27/2019 Mod10 Control Chart

1/70

Basic Tools for Process Improvement

Module 10

CONTROL CHAR


2/70


What is a Control Chart?

A control chart is a statistical tool used to distinguish between variation in a

resulting from common causes and variation resulting from special causes.presents a graphic display of process stability or instability over time (Viewg

Every process has variation. Some variation may be the result of causes w

not normally present in the process. This could be special cause variatio

variation is simply the result of numerous, ever-present differences in the p

This is comm on cause variation. Control Charts differentiate between th

types of variation.

One goal of using a Control Chart is to achieve and maintain process stab

Process stability is defined as a state in which a process has displayed a cedegree of consistency in the past and is expected to continue to do so in th

This consistency is characterized by a stream of data falling within control

based on plus or minus 3 standard deviations (3 sigma) of the centerl6, p. 82]. We will discuss methods for calculating 3 sigma limits later in this

NOTE: Contro l lim its represent the limits of variation that should be expe

a process in a state of statistical control. When a process is in statistical covariation is the result of common causes that effect the entire production in

way. Control limits should not be confused with specification limits, whic

represent the desired process performance.

Why should teams use Control Charts?

A stable process is one that is consistent over time with respect to the centspread of the data. Control Charts help you monitor the behavior of your p

determine whether it is stable. Like Run Charts, they display data in the t i

sequence in which they occurred. However, Control Charts are more e

that Run Charts in assessing and achieving process stability.

Your team will benefit from using a Control Chart when you want to (Viewgr

Monitor process variation over time.


3/70

Why Use Control Charts?

Monitor process variation over time

Differentiate between special cause and

common cause variation

Assess effectiveness of changes

CONTROL CHART VIEW

What Is a Control Chart?

A statistical tool used to distinguish

between process variation resulting

from common causes and variation

resulting from special causes.



4/70


What are the types of Control Charts?

There are two main categories of Control Charts, those that display attribut

and those that display variables data.

Attribute Data: This category of Control Chart displays data that re

counting the number of occurrences or items in a single category of

items or occurrences. These count data may be expressed as pasyes/no, or presence/absence of a defect.

Variables Data: This category of Control Chart displays values res

from the measurement of a continuous variable. Examples of variabare elapsed time, temperature, and radiation dose.

While these two categories encompass a number of different types of Cont(Viewgraph 3), there are three types that will work for the majority of the da

cases you will encounter. In this module, we will study the construction andapplication in these three types of Control Charts:

X-Bar and R Chart

Individual X and Moving Range Chart for Variables Data

Individual X and Moving Range Chart for Attribute Data

Viewgraph 4 provides a decision tree to help you determine when to use thtypes of Control Charts.

In this module, we will study only the Individual X and Moving Range Contro

for handling attribute data, although there are several others that could be uas the np, p, c, and u charts. These other charts require an understanding probability distribution theory and specific control limit calculation formulas w

not be covered here. To avoid the possibility of generating faulty results byimproperly using these charts, we recommend that you stick with the IndividMoving Range chart for attribute data.

The following six types of charts will not be covered in this module:


5/70

CONTROL CHART VIEW

What Are the Control Chart Type

Chart types studied in this moduleX-Bar and R Chart

Individual X and Moving Range Chart

- For Variables Data

- For Attribute Data

Other Control Chart types:X-Bar and S Chart u Chart

Median X and R Chart p Chartc Chart np Chart

Control Chart Decision Tree

Are you

charting

attribute

data?

YES

NO YES

NO

Data are

variables

data

Is sample

size

equal to

1?

Use X

chart

variab

dat

For sample size

between 2 and

Use XmR

chart for



6/70


What are the elements of a Cont rol Chart?

Each Control Chart actually consists oftwo graphs, an upperand a lower

are described below underplotting areas. A Control Chart is made up of ei

elements. The first three are identified in Viewgraphs 5; the other five in Vie6.

1. Title. The title briefly describes the information which is displayed.

2. Legend. This is information on how and when the data were collect

3. Data Collection Section. The counts or measurements are record

data collection section of the Control Chart prior to being graphed.

4. Plotting A reas. A Control Chart has two areasan upper graph an

graphwhere the data is plotted.

a. The upper graph plots either the individual values, in the case o

Individual X and Moving Range chart, or the average (mean valu

sample or subgroup in the case of an X-Bar and R chart.

b. The lower graph plots the moving range for Individual X and Mo

Range charts, or the range of values found in the subgroups for X

R charts.

5. Vertical or Y-Axis. This axis reflects the magnitude of the data col

The Y-axis shows the scale of the measurement forvariables dat

count (frequency) or percentage of occurrence of an event for

data.

6. Horizontal or X-Axis . This axis displays the chronological order in

data were collected.

7. Control Limits. Control limits are set at a distance of 3 sigma abov

sigma below the centerline [Ref. 6, pp. 60-61]. They indicate variatiothe centerline and are calculated by using the actual values plotted o

Control Chart graphs.


7/70

CONTROL CHART VIEW

Elements of a Control Chart

M

E

A

S

U

RE

M

E

N

T

S

Date

Title: ____________________________________ Legend:_____________________

Average

Range

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14

21

3

Elements of a Control Chart

.

. .

..

.

5

4a

8

6

7

4b



8/70

xx

1x

2x

3...x

n

n

Where: x The average of the measurements within each subgroupxi

The individual measurements within a subgroup

n The number of measurements within a subgroup

Subgroup 1 2 3 4 5 6 7 8 9X1 15.3 14.4 15.3 15.0 15.3 14.9 15.6 14.0 14.0

X2 14.9 15.5 15.1 14.8 16.4 15.3 16.4 15.8 15.2

X3 15.0 14.8 15.3 16.0 17.2 14.9 15.3 16.4 13.6

X4 15.2 15.6 18.5 15.6 15.5 16.5 15.3 16.4 15.0

X5 16.4 14.9 14.9 15.4 15.5 15.1 15.0 15.3 15.0

Average: 15 36 15 04 15 82 15 36 15 98 15 34 15 52 15 58 14 56


What are the steps for calculating and p lotting an

X-Bar and R Control Chart for Variables Data?

The X-Bar (arithmetic mean) and R (range) Control Chart is used with var

data when subgroup or samp le size is between 2 and 15. The steps

constructing this type of Control Chart are:

Step 1 - Determine the data to be collected. Decide what questions ab

process you plan to answer. Refer to the Data Collection module for infon how this is done.

Step 2 - Collect and enter the data by subgroup. A subgroup is made

variables data that represent a characteristic of a product produced by aThe sample size relates to how large the subgroups are. Enter the indivsubgroup measurements in time sequence in the portion of the data col

section of the Control Chart labeled MEASUREMENTS (Viewgraph 7).

STEP 3 - Calculate and enter the average for each subgroup. Use th

below to calculate the average (mean) for each subgroup and enter it on

labeled Average in the data collection section (Viewgraph 8).

Average Example


9/70

CONTROL CHART VIEW

Constructing an X-Bar & R ChartStep 2 - Collect and enter data by subgrou

A

V

ER

A

GE

Date

Title: ____________________________________ Legend: ____________________

1 Feb 2 Feb 5 Feb 6 Feb 7 Feb 8 Feb3 Feb 4 Feb 9 Feb

M

E

ASU

R

EM

E

NT

S

Average

Range

1

2

4

5

6

1 2 3 4 5 6 7 8 9

14.9 16.4 15.315.5 14.815.1 15.216.4 15.8

3 15.0 14.9 15.314.8 16.015.3 17.2 13.616.4

15.2 15.6 15.618.5 15.5 15.316.5 15.016.4

16.4 14.9 15.414.9 15.115.5 15.0 15.015.3

15.3 15.3 14.9 15.614.4 15.015.3 14.014.0

Enter data by subg r

in time sequenc

Constructing an X-Bar & R ChartStep 3 - Calculate and enter subgroup avera

Date

Title: ____________________________________ Legend: ____________________


M

E

AS

U

RE

M

E

NT

S

Average

Range

1

2

4

5

6

14.9 16.4 15.315.5 14.815.1 15.216.4 15.8

3 15.0 14.9 15.314.8 16.015.3 17.2 13.616.4

15.2 15.6 15.618.5 15.5 15.316.5 15.016.4

16.4 14.9 15.414.9 15.115.5 15.0 15.015.3

15.3 15.3 14.9 15.614.4 15.015.3 14.014.0

15.36 15.04 15.82 15.36 15.34 15.58 14.5615.98 15.52



10/70

RANGE (Largest Value in each Subgroup) (Smallest Value in each Sub

Subgroup 1 2 3 4 5 6 7 8 9X1 15.3 14.4 15.3 15.0 15.3 14.9 15.6 14.0 14.0

X2 14.9 15.5 15.1 14.8 16.4 15.3 16.4 15.8 15.2X3 15.0 14.8 15.3 16.0 17.2 14.9 15.3 16.4 13.6

X4 15.2 15.6 18.5 15.6 15.5 16.5 15.3 16.4 15.0

X5 16.4 14.9 14.9 15.4 15.5 15.1 15.0 15.3 15.0

Average: 15.36 15.04 15.82 15.36 15.98 15.34 15.52 15.58 14.56

Range: 1.5 1.2 3.6 1.2 1.9 1.6 1.4 2.4 1.6

xx

1x

2x

3...x

k

k

Where: x The grand mean of all the individual subgroup averag

x The average for each subgroupk The number of subgroups

x15.36 15.04 15.82 15.36 15.98 15.34 15.52 15.58 14.56 1


Step 4 - Calculate and enter the range for each subgroup. Use the fo

formula to calculate the range (R) for each subgroup. Enter the range fsubgroup on the line labeled Range in the data collection section (Viewg

Range Example

The rest of the steps are listed in Viewgraph 10.

Step 5 - Calculate the grand mean of the subg roup s average. The

mean of the subgroups average (X-Bar) becomes the centerline for theplot.

Grand Mean Example


11/70

Constructing an X-Bar & R Cha

Step 5 - Calculate grand mean

Step 6 - Calculate average of subgroup ran

Step 7 - Calculate UCL and LCL for subgro

averages

Step 8 - Calculate UCL for ranges

CONTROL CHART VIEW

Constructing an X-Bar & R ChartStep 4 - Calculate and enter subgroup rang

A

V

ER

A

GE

Date

Title: ____________________________________ Legend: ____________________


M

E

ASU

R

EM

E

NT

S

Average

Range

1

2

4

5

6

1 2 3 4 5 6 7 8 9

14.9 16.4 15.315.5 14.815.1 15.216.4 15.8

3 15.0 14.9 15.314.8 16.015.3 17.2 13.616.4

15.2 15.6 15.618.5 15.5 15.316.5 15.016.4

16.4 14.9 15.414.9 15.115.5 15.0 15.015.3

15.3 15.3 14.9 15.614.4 15.015.3 14.014.0

Enter the range foeach subg roup

15.36 15.04 15.82 15.36 15.34 15.58 14.5615.98 15.52

1.5 1.2 3.6 1.2 1.9 1.6 1.4 2.4 1.6



12/70

RR1 R2 R3 ...Rk

kWhere: Ri The individual range for each subgroup

R The average of the ranges for all subgroupsk The number of subgroups

R1.5 1.2 3.6 1.2 1.9 1.6 1.4 2.4 1.6

9

16.4

91.8

UCLX X A2RLCLX X A2 R


Step 6 - Calculate the average of the subgroup ranges. The average

subgroups becomes the centerline for the lower plotting area.

Average o f Ranges Example

Step 7 - Calculate the upper contro l lim it (UCL) and lower contro l lifor the averages of the subg roups. At this point, your chart will look

Run Chart. Now, however, the uniqueness of the Control Chart becomeas you calculate the control limits. Control limits define the parameters

determining whether a process is in statistical control. To find the X-Balimits, use the following formula:

NOTE: Constants, based on the subgroup size (n), are used in determini

limits for variables charts. You can learn more about constants in Tools an

for the Improvement of Quality [Ref. 3].

Use the following constants (A ) in the computation [Ref. 3, Table 8]:2

n A n A n A2 2 2

2 1 880 7 0 419 12 0 266


13/70

UCLX

X A2R (15.40) (0.577)(1.8) 16.4386

LCLX X A2R (15.40) (0.577)(1.8) 14.3614

UCLR

D4

R

LCLR D3 R (for subgroups 7)

UCLR D4R (2.114)(1.8) 3.8052


Upper and Lower Control Lim its Example

Step 8 - Calculate the upper con trol limit fo r the ranges. When the

or sample size (n) is less than 7, there is no lower control limit. To find

control limit for the ranges, use the formula:

Use the following constants (D ) in the computation [Ref. 3, Table 8]:4

n D n D n D4 4 4

2 3.267 7 1.924 12 1.717

3 2.574 8 1.864 13 1.693

4 2.282 9 1.816 14 1.672

5 2.114 10 1.777 15 1.653

6 2.004 11 1.744

Example


14/70


Step 9 - Select the scales and plot the contro l lim its, centerline, and

poin ts, in each plotti ng area. The scales must be determined before

points and centerline can be plotted. Once the upper and lower control

have been computed, the easiest way to select the scales is to have thedata take up approximately 60 percent of the vertical (Y) axis. The scal

the upper and lower plotting areas should allow for future high or low oucontrol data points.

Plot each subgroup average as an individual data point in the upper

area. Plot individual range data points in the lower plotting area (Vie11).

Step 10 - Provide the appropriate docum entation. Each Control Char

be labeled with who, what, when, where, why, and how information to dwhere the data originated, when it was collected, who collected it, any id

equipment or work groups, sample size, and all the other things necessunderstanding and interpreting it. It is important that the legend include information that clarifies what the data describe.

When should we use an Individual X and Moving Ran(XmR) Control Chart?

You can use Individual X and Moving Range (XmR) Control Charts to assevariables and attribute data. XmR charts reflect data that do not lend themforming subgroups with more than one measurement. You might want to u

type of Control Chart if, for example, a process repeats itself infrequently, oappears to operate differently at different times. If that is the case, groupingmight mask the effects of such differences. You can avoid this problem by

XmR chart whenever there is no rational basis for grouping the data.

What conditions must we satisfy to use an XmR Con

Chart for attribute data?

The only condition that needs to be checked before using the XmR Control

that the average count per sample IS GREATER THAN ONE.


15/70

CONTROL CHART VIEW

Constructing an X-Bar & R ChartStep 9 - Select scales and plot

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

A

V

E

R

A

GE

R

A

N

GE

3.0

2.0

1.0

0

16.0

15.5

15.0

14.5

14.0

16.5

.



16/70

mRi Xi 1 X i

Where: Xi

Is an individual value

Xi 1 Is the next sequential value following Xi

Note: The brackets ( ) refer to the absolute valueof the numbers contained inside the bracket. In thisformula, the difference is always a positive number.


What are the steps for calculating and plott ing an Xm

Control Chart?

Step 1 - Determine the data to be collected. Decide what questions ab

process you plan to answer.

Step 2 - Collect and enter the indiv idual measurements . Enter the in

measurements in time sequence on the line labeled Individual X in the dcollection section of the Control Chart (Viewgraph 12). These measure

be plotted as individual data points in the upper plotting area.

STEP 3 - Calculate and enter the moving ranges. Use the following fo

calculate the moving ranges between successive data entries. Enter th

line labeled Moving R in the data collection section. The moving range be plotted as individual data points in the lower plotting area (Viewgraph

Example


17/70

CONTROL CHART VIEW

Constructing an XmR ChartStep 2 - Collect and enter individual measurem

Title: ____________________________________ Legend: ___________________

Date 1 Apr 2 Apr 5 Apr 6 Apr 7 Apr 8 Apr 3 Apr 4 Apr 9 Apr 10Apr

Moving R

1 2 3 4 5 6 7 8 9 10

Individual X 19 22 16 18 19 23 18 15 19 18

Enter individual

measurements in

time sequence

I

NDI

V

I

D

U

AL

X

Constructing an XmR ChartStep 3 - Calculate and enter moving rangesTitle: ____________________________________ Legend: ___________________

Date 1 Apr 2 Apr 5 Apr 6 Apr 7 Apr 8 Apr 3 Apr 4 Apr 9 Apr 10Apr

Moving R

1 2 3 4 5 6 7 8 9 10

Individual X 19 22 16 18 19 23 18 15 19 18

3 6 2 1 4 5 3 4 1

IN

D



18/70

xx

1x

2x

3...x

n

k

Where: x The average of the individual measurementsx

iAn individual measurement

k The number of subgroups of one

X19 22 16 18 19 23 18 15 19 18

10

169

1016.9

mRmR

1mR

2mR

3...mR

n

k 1

Where: mR The average of all the Individual Moving RangesmRn The Individual Moving Range measurements

k The number of subgroups of one

mR3 6 2 1 4 5 3 4 1

(10 1)

29

93.2


Example

Steps 4 through 9 are outlined in Viewgraph 14.

Step 4 - Calculate the overall average of the indiv idual data poin ts.average of the Individual-X data becomes the centerline for the upper pl

Step 5 - Calculate the average of the moving ranges. The average of

moving ranges becomes the centerline for the lower plotting area.

Average of Moving Ranges Exam ple

Step 6 - Calculate the upper and lower cont rol lim its for the individ

values. The calculation will compute the upper and lower control limits

upper plotting area To find these control limits use the formula:


19/70

CONTROL CHART VIEW

Constructing an XmR Chart

Step 4 - Calculate average of data points

Step 5 - Calculate average of moving range

Step 6 - Calculate UCL and LCL for individu

Step 7 - Calculate UCL for ranges

Step 8 - Select scales and plot

Step 9 - Document the chart



20/70

UCLX

X (2.66)mR (16.9) (2.66)(3.2) 25.41

LCLX X (2.66)mR (16.9) (2.66)(3.2) 8.39

UCLmR

(3.268)mR

UCLmR NONE

UCLmR (3.268)mR (3.268)(3.2) 10.45


You should use the constant equal to 2.66 in both formulas when you cothe upper and lower control limits for the Individual X values.

Upper and Lower Control Lim its Example

Step 7 - Calculate the upper control lim it for the ranges. This calculacompute the upper control limit for the lower plotting area. There is no locontrol limit. To find the upper control limit for the ranges, use the formu

You should use the constant equal to 3.268 in the formula when you comupper control limit for the moving range data.

Example

Step 8 - Select the scales and plot the data points and centerline in

plotting area. Before the data points and centerline can be plotted, the

must first be determined. Once the upper and lower control limits have computed, the easiest way to select the scales is to have the current sp

the control limits take up approximately 60 percent of the vertical (Y) axscales for both the upper and lower plotting areas should allow for future

low out-of-control data points.

Plot each Individual X value as an individual data point in the upper p

area. Plot mov ing range values in the lower plotting area (Viewgra

St 9 P id th i t d t ti E h C t l Ch t


21/70

CONTROL CHART VIEW

Constructing an XmR ChartStep 8 - Select scales and plot

.

.

.

.

.

.

.

.

.

.

I

N

D

I

V

I

D

U

A

L

X

R

A

N

GE0

.

21

18

15

12

9

24

6

27

.

.

.

.

.

. .

9

6

3

12

.



22/70

( T o t a l = 1 2 )

2

34

5

5

6

7

7

M E D I A N = 6 .5

( T o t a l = 1 1 )

2

34

5

5

6

7

7

M E D I A N = 6 .0


EVEN ODD

NOTE: If you are working with attribute data, continue through step

12a, and 12b (Viewgraph 16).

Step 10 - Check for inflated control l imits. You should analyze your X

Control Chart for inflated control limits. When either of the following conexists (Viewgraph 17), the control limits are said to be inflated, and you

recalculate them:

If any point is outside of the upper control limit for the moving range

If two-thirds or more of the moving range values are below the avera

moving ranges computed in Step 5.

Step 11 - If the control l imit s are inflated, calculate 3.144 times the

moving range. For example, if the median moving range is equal to 6,

(3.144)(6) = 18.864

The centerline for the lower plotting area is now the median of all the va

the mean) when they are listed from smallest to largest. Review the dismedian and centerline in the Run Chart module for further clarification.

When there is an odd number of values, the median is the middle va

When there is an even number of values, average the two middle va

obtain the median.

Example


23/70

Constructing an XmR ChartStep 10 - Check for inflated control limits

UCLmR

mR

Any po int is above

upper control limit of

moving range

CONTROL CHART VIEW

Constructing an XmR Chart

Step 10 - Check forinflated control limits

Step 11 - If inflated, calculate 3.144 times

median mRStep 12a - Do not recompute if 3.144 time

median mR is greater than 2.66

times average of moving range

Step 12b - Otherwise, recompute all contro

limits and centerlines



24/70


Step 12a - Do not compute new lim its if t he product of 3.144 times

median mov ing range value is greater than the product of 2.66

average of the mov ing ranges.

Step 12b - Recompute all of the control limits and centerlines for bo

upper and lower plot ting areas if the product o f 3.144 times the

mov ing range value is less than the product o f 2.66 times the a

the moving range. The new limits will be based on the formulas found

Viewgraph 18.

These new limits must be redrawn on the corresponding charts before yfor signals of special causes. The old control limits and centerlines are

any further assessment of the data.


25/70

CONTROL CHART VIEW

Upper Plot

UCLX = X + (3.144) (Median Moving Range)

LCLX = X - (3.144) (Median Moving Range)CenterlineX = X

Lower Plot

UCLmR = (3.865) (Median Moving Range)

LCLmR = None

CenterlinemR = Median Moving Range

Step 12b - Constructing an XmChart



26/70


What do we need to know to interpret Control Charts

Process stability is reflected in the relatively constant variation exhibited in

Charts. Basically, the data fall within a band bounded by the control limits. process is stable, the likelihood of a point falling outside this band is so sma

such an occurrence is taken as a signal of a special cause of variation

words, something abnormal is occurring within your process. However, ev

all the points fall inside the control limits, special cause variation may be at The presence of unusual patterns can be evidence that your process is not

statistical control. Such patterns are more likely to occur when one or morecauses is present.

Control Charts are based on control limits which are 3 standard deviations

away from the centerline. You should resist the urge to narrow these limits

hope of identifying special causes earlier. Experience has shown that l imit

on less than plus and minus 3 sigma may lead to false assumption

special causes operating in a process [Ref. 6, p. 82]. In other words, usin

limits which are less than 3 sigma from the centerline may trigger a hunt forcauses when the process is already stable.

The three standard deviations are sometimes identified by zones. Each zodividing line is exactly one-third the distance from the centerline to either thcontrol limit or the lower control limit (Viewgraph 19).

Zone A is defined as the area between 2 and 3 standard deviations centerline on both the plus and minus sides of the centerline.

Zone B is defined as the area between 1 and 2 standard deviations

centerline on both sides of the centerline.

Zone C is defined as the area between the centerline and 1 standar

deviation from the centerline, on both sides of the centerline.

There are two basic sets of rules for interpreting Control Charts:

Rules for interpreting X-Bar and R Control Charts.


27/70

CONTROL CHART VIEW

Control Chart Zones

1/3 distfro

Cente

to Co

Lim

LCL

Centerline

ZONE A

ZONE B

ZONE C

ZONE C

ZONE B

ZONE A

UCL


B i T l f P I t


28/70


What are the rules for interpreting X-Bar and R Chart

When a special cause is affecting the data, the nonrandom patterns display

Control Chart will be fairly obvious. The key to these rules is recognizing thserve as a signal for when to investigate what occurred in the process.

When you are interpreting X-Bar and R Control Charts, you should apply thfollowing set of rules:

RULE 1 (Viewgraph 20): Whenevera single point falls outside the 3 scontrol l imits, a lack of control is indicated. Since the probability of th

happening is rather small, it is very likely not due to chance.

RULE 2 (Viewgraph 21): Whenever at least 2 out of 3 successive value

the same side of the centerline and more than 2 sigma units away from centerline (in Zone A or beyond), a lack of control is indicated. Note tha

point can be on either side of the centerline.



29/70

CONTROL CHART VIEW

Rule 1 - Interpreting X-Bar & R Ch

.

.

.

.

.

.

.

.

.

Centerline

UCL

LCL

ZO

Out of Limits

ZO

ZO

ZO

ZO

ZO

.

.

.

.

UCL

Centerline

ZON

ZON

ZON

ZON

Rule 2 - Interpreting X-Bar & R Cha




30/70


RULE 3 (Viewgraph 22): Whenever at least 4 out of 5 successivevalue

the same side of the centerline and more than one sigma unit away from

centerline (in Zones A or B or beyond), a lack of control is indicated. Nofifth point can be on either side of the centerline.

RULE 4 (Viewgraph 23): Whenever at least 8 successive values fall on

side of the centerline, a lack of control is indicated.

What are the rules for interpreting XmR Control Char

To interpret XmR Control Charts, you have to apply a set of rules for interp

X part of the chart, and a further single rule for the mR part of the chart.

RULES FOR INTERPRETING THE X-PORTION of XmR Control C

Apply the four rules discussed above, EXCEPT apply them only to the

plotting area graph .

RULE FOR INTERPRETING THE mR PORTION of XmR Control C

attribute data: Rule 1 is the only rule used to assess signals of speci

in the lower plott ing area graph. Therefore, you dont need to identify

on the moving range portion of an XmR chart.



31/70

CONTROL CHART VIEW

.

.

. .

UCL

LCL

Centerline

4 out of 5

successive valuesin Zones A & B

ZON

ZON

ZON

ZON

ZON

ZON


UCL

Centerline


ZO

ZO

ZO

ZO




32/70


When should we change the control l imits?

There are only three situations in which it is appropriate to change the cont

When removing out-of-control data points. When a special cau

been identified and removed while you are working to achieve procestability, you may want to delete the data points affected by special c

and use the remaining data to compute new control limits.

When replacing trial limits . When a process has just started up, changed, you may want to calculate control limits using only the limitavailable. These limits are usually called trial control limits. You can

new limits every time you add new data. Once you have 20 or 30 gror 5 measurements without a signal, you can use the limits to monitoperformance. You dont need to recalculate the limits again unless

fundamental changes are made to the process.

When there are changes in the process . When there are indicatyour process has changed, it is necessary to recompute the control

based on data collected since the change occurred. Some examplechanges are the application of new or modified procedures, the use machines, the overhaul of existing machines, and the introduction of

suppliers of critical input materials.

How can we practice what we've learned?

The exercises on the following pages will help you practice constructing aninterpreting both X-Bar and R and XmR Control Charts. Answer keys follow

exercises so that you can check your answers, your calculations, and the gcreate for the upper and lower plotting areas.



33/70

p

EXERCISE 1: A team collected the variables data recorded in the table

1 2 3 4 5 6 7 8 9 10 11 12 13 1

X1 6 2 5 3 2 5 4 7 2 5 3 6 4 5

X2 5 7 6 6 8 4 6 4 3 5 1 4 3 4

X3 2 9 4 6 3 8 3 4 7 2 6 2 6 6

X4 7 3 2 7 5 4 6 5 1 6 5 2 6 2

Use these data to answ er the following questions and plo t a Cont

1. What type of Control Chart would you use with these data?

2. Why?

3. What are the values of X-Bar for each subgroup?

4. What are the values of the ranges for each subgroup?

5. What is the grand mean for the X-Bar data?

6. What is the average of the range values?

7. Compute the values for the upper and lower control limits for both the

lower plotting areas.

8. Plot the Control Chart.

9. Are there any signals of special cause variation? If so, what rule did yidentify the signal?



34/70

p

EXERCISE 1 ANSWER KEY:

1. X-Bar and R.

2. There is more than one measurement within each subgroup.

3. Refer to Viewgraph 24.


5. Grand Mean of X = 4.52.

6. Average of R = 4.38.

7. UCL = 4.52 + (0.729) (4.38) = 7.71.X

LCL = 4.52 - (0.729) (4.38) = 1.33.X

UCL = (2.282) (4.38) = 10.0.R

LCL = 0.R


9. No.



35/70

CALCULATIONS



36/70

CONTROL CHART VIEWG

EXERCISE 1

Values of X-Bar and Ranges

X1

X 2

X3

42 4 6 5 6 27 57 3 5 6 2 31

8 3 4 6 64 6 62 9 3 2 2 77

4 6 4 3 46 6 15 7 8 5 4 43

3 5 36 2 5 2 4 7 5 6 4 5 32

3 6 7 8 9 13 14 154 111 2 5 10 12

R

3.55.5 4.55.0 5.3 4.3 5.3 4.8 5.0 3.3 4.5 3.8 4.8 4.3 4.3

5 7 4 4 6 3 3 6 4 5 4 3 4 4

X

X-Bar

4

ZONE A

ZONE B

ZONE C

ZONE C

ZONE B

ZONE A

EXERCISE 1X-Bar & R Control Chart

A

V

E

R

AG

E

R ZONE A

.

.

.

.

.

.

.

..

.

.

.

.

.

.

.

7

6

5

4

3

2

1

9



37/70

EXERCISE 2: A team collected the dated shown in the chart below.

Date 1 2 3 4 5 6 7 8

X

Value16 20 21 8 28 24 19 16

Date 11 12 13 14 15 16 17 18

X

Value19 22 26 19 15 21 17 22

Use these data to answ er the following questions and plo t a Cont

1. What are the values of the moving ranges?

2. What is the average for the individual X data?

3. What is the average of the moving range data?

4. Compute the values for the upper and lower control limits for both tand lower plotting areas.

5. Plot the Control Chart.

6. Are the control limits inflated? How did you determine your answe

7. If the control limits are inflated, what elements of the Control Chart have to recompute?

8. If the original control limits were inflated, what are the new values f

and lower control limits and centerlines?

9. If the original limits were inflated, replot the Control Chart using the

information.

10 Aft h ki f i fl t d li it th i l f i l



38/70

EXERCISE 2 ANSWER KEY:


2. 19.2.

3. 5.5.

4. UCL = 33.8.X

LCL = 4.6.X

UCL =18.0.mR

LCL = 0.mR


6. Yes; one point out of control, and 2/3 of all points below the centerlin

7. All control limits and the centerline for the lower chart. The median vbe used in the recomputation rather than the average.

8. UCL = 31.8.X

LCL = 6.6.X

UCL = 15.5.mR

LCL = 0.mR

Centerline = 4 (median value).mR


10. Yes; the same point on the mR chart is out of control. Rule 1.



39/70

CONTROL CHART VIEW

EXERCISE 2

Values of Moving Ranges

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Date

816 20 21 28 24 19 16 17 24 19 22 26 19 15 21 17 22X Values

134 1 20 4 5 3 1 7 5 3 4 7 4 6 4 5mR

EXERCISE 2XmR Control Chart

..

..

.

.

.

.

.

.

.

.

. .

.

. .

.

.

IN

D

IV

I

D

UA

L

X

35

30

25

20

15

10

05

.

M

O



40/70

CONTROL CHART VIEWG

EXERCISE 2XmR Control Chart Revised for Inflated Lim

Z

Z

Z

Z

Z

Z

I

ND

I

V

I

DU

A

L

X

35

30

25

20

15

10

05

15

10

05

0

M

O

V

IN

G

R

A

NG

E

.

Old Control Limits New Control Limits

Note: Solid lines represent the grid used in this module; light dashed lines divide the zones in the upper p

Old Centerline New Centerline

Old Control Limit

New Control Limit



41/70

REFERENCES:

1. Department of the Navy (November 1992). Fundamentals of Total QuLeadership (Instructor Guide), pp, 3-39 - 3-66 and 6-57 - 6-62. San DNavy Personnel Research and Development Center.

2. Department of the Navy (September 1993). Systems Approach to ProImprovement (Instructor Guide), Lessons 8 and 9. San Diego, CA: O

Quality Leadership Office and Navy Personnel Research and Develop

Center.

3. Gitlow, H., Gitlow, S., Oppenheim, A., Oppenheim, R. (1989). Tools afor the Improvement of Quality. Homewood, IL: Richard D. Irwin, Inc.

4. U.S. Air Force (Undated). Process Improvement Guide - Total QualityTeams and Individuals, pp. 61 - 81. Air Force Electronic Systems CenForce Materiel Command.

5. Wheeler, D.J. (1993). Understanding Variation - The Key to ManagingKnoxville, TN: SPC Press.

6. Wheeler, D.J., & Chambers, D.S. (1992). Understanding Statistical P

Control (2nd Ed.). Knoxville, TN: SPC Press.



42/70

C


43/70

CONTROLCHART

VIEWGRAPH

1

Wha

tIsaCo

ntrolCh

art?

Astatisticaltoolusedtodistinguish

between

processvariationr

esulting

fromcom

moncau

sesandv

ariation

resulting

fromspe

cialcause

s.

C


44/70

CONTROLCHART

VIEWGRAPH

2

Why

UseControlCharts?

Monitorp

rocessvar

iationover

time

Differentiatebetwee

nspecialcauseand

common

causevariation

Assesse

ffectivenes

sofchange

s

Commun

icateproce

ssperform

ance

C


45/70

CONTROLCHART

VIEWGRAPH

3

W

hatAre

theCon

trolCha

rtTypes

?

Charttyp

esstudiedinthis

module:

X-Bara

ndR

Chart

IndividualXandMovingRangeC

hart

-ForVariablesData

-ForAttributeData

Othe

rControlC

harttypes

:

X-Baran

dSChart

uChart

MedianX

andR

Chart

pChart

cChart

npChart

CO


46/70

ONTROLCHART

VIEWGRAPH

4

ControlChart

DecisionTree

A

reyou

charting

attribute

data?

YES

NO

YES

NO

Dataare

variables

data

Issample

size

equalto

1?

UseXmR

chartfor

variables

data

Forsamplesize

between2and

15,useX-Bar

andRChart

Us

eXmR

chartfor

attribute

data

CO


47/70

ONTROLC

HART

VIEWGRAPH

5

Eleme

ntsofa

Control

Chart

MEASUREMENTSDate

Title:_________________

__________________

_Legend:_________

_______________

Average

Range 123456

1

2

3

4

5

6

7

8

9

10

11

12

13

14

2

1

3

CO


48/70

ONTROLC

HART

VIEWGRAPH6

Eleme

ntsofa

Control

Chart

0

.

..

.

.. ..

...5

4a

8

6

7

.

5

6

7

4b

8

CO


49/70

ONTROLC

HART

VIEWGRAPH7

Constru

ctingan

X-Bar&RChart

Step2-Collectanden

terdataby

subgroup

AVERAGEDate

Title:_________________

__________________

_Legend:_________

______________

1Feb2Feb

5Feb6Feb7Feb

8

Feb

3Feb

4Feb

9Feb

MEASUREMENTSAverage

Range 12456

1

2

3

4

5

6

7

8

9

14.9

16.4

15.3

15.5

14

.8

15.1

15.2

16.4

15.8

3

15.0

14.9

15.3

14.8

16

.0

15.3

17.2

13.6

16.4

15.2

15.6

15

.6

18.5

15.5

15.3

16.5

15.0

16.4

16.4

14.9

15

.4

14.9

15.1

15.5

15.0

15.0

15.3

15.3

15.3

14.9

15.6

14.4

15

.0

15.3

14.0

14.0

Enterdatabysubgroup

intim

esequence


50/70

CON


51/70

NTROLC

HART

VIEWGRAPH9

Constru

ctingan

X-Bar&RChart

S

tep4-Calc

ulateande

ntersubgro

upranges

AVERAGEDate

Title:

___________________________________

_Legend:_________

______________

1Feb2Feb

5Feb6Feb7Feb

8

Feb

3Feb

4F

eb

9Feb

MEASUREMENTSAverag

e

Range 12456

1

2

3

4

5

6

7

8

9

14.9

16.4

15.3

15.5

14.8

15.1

15.2

16.4

15.8

3

15.0

14.9

15.3

14.8

16.0

15.3

17.2

13.6

16.4

15.2

15.6

15.6

18.5

15.5

15.3

16.5

15.0

16.4

16.4

14.9

15.4

14.9

15.1

15.5

15.0

15.0

15.3

15.3

15.3

14.9

15.6

14.4

15.0

15.3

14.0

14.0

Enter

therangefor

eac

hsubgroup

15.36

15.0415.8215.36

15.34

15.5814.56

15.98

15.52

1.5

1.2

3.6

1

.2

1.9

1.6

1.4

2.4

1.6

CON


52/70

NTROLC

HART

VIEWGRAPH10

Constru

ctingan

X-Bar&

RChart

Step5

-Calc

ulategrandmean

Step6

-Calc

ulateavera

geofsubg

rouprange

s

Step7

-Calc

ulateUCL

andLCLfo

rsubgroup

averages

Step8

-Calc

ulateUCL

forranges

Step9

-Sele

ctscalesa

ndplot

Step10-Doc

umentthechart

CON


53/70

NTROLC

HART

VIEWGRAPH11

Constru

ctingan

X-Bar&RChart

Step

9-Selects

calesandp

lot

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

AVERAGERANGE

3.0

2.0

1.00

16.0

15.5

15.0

14.5

14.0

16.5

.

CON S


54/70

TROLC

HART

VIEWGRAPH12

Cons

tructinganXmRC

hart

Step2-Collec

tandenterindividualmeasurements

Title

:__________________________________

__Legend:________

_____________

Date

1Apr2Ap

r

5Apr6A

pr7Apr8Apr

3Apr4Apr

9Apr10Apr

MovingR

1

2

3

4

5

6

7

8

9

1

0

IndividualX

19

22

16

18

19

2

3

18

15

19

1

8

Ente

rindividual

meas

urementsin

time

sequence

INDIVIDUALX

CONT


55/70

TROLC

HART

VIEWGRAPH13

Cons

tructinganXmRC

hart

Step3-Ca

lculateand

entermovin

granges

Title

:__________________________________

__Legend:________

_____________

Date

1Apr2Ap

r

5Apr6A

pr7Apr8Apr

3Apr4Apr

9Apr10Apr

MovingR

1

2

3

4

5

6

7

8

9

1

0

IndividualX

19

22

16

18

19

23

18

15

19

1

8

Enterthe

movingranges

3

6

2

1

4

5

3

4

1

INDIVIDUALX

CONT

SSSSSS


56/70

TROLC

HART

VIEWGRAPH14

Const

ructinganXmR

Chart

Ste

p4

-Calculateavera

geofdatapoints

Ste

p5

-Calculateavera

geofmovin

granges

Ste

p6

-CalculateUCLa

ndLCLfor

individualX

Ste

p7

-CalculateUCLf

orranges

Ste

p8

-Selectscalesan

dplot

Ste

p9

-Docu

mentthec

hart

CONT


57/70

TROLC

HART

VIEWGRAPH1

5

Cons

tructinganXmRC

hart

Step

8-Selects

calesandp

lot

.

.

.

.

.

.

.

.

.

.

INDIVIDUALXRANGE

0

.

21

18

15

129

246

27

.

.

.

.

.

.

.

96312

.

CONT


58/70

ROLC

HART

VIEWGRAPH1

6

Const

ructinganXmR

Chart

S

tep10

-C

heckforinflatedcontrollimits

S

tep11

-Ifinflated,calculate3.1

44times

m

edianmR

S

tep12a-D

onotrecomputeif3.

144times

m

edianmR

isgreatert

han2.66

timesaverageofmovin

granges

S

tep12b-O

therwise,recomputeallcontrol

limitsandce

nterlines

CONTR


59/70

ROLC

HART

VIEWGRAPH1

7

Constructinga

nXmRC

hart

Step10-C

heckforin

flatedcontr

ollimits

U

CLmR

mR0

Anypointisabove

uppercontrollimitof

moving

range

2/3ormoreofdata

pointsare

belowaverageofmovingrange

(13of19datapoin

ts=68%)

CONTR


60/70

ROLC

HART

VIEWGRAPH1

8

Upper

Plot

UCLX=

X+

(3.144)(

MedianMoving

Range)

LCLX

=

X

-(3.144)(

MedianMoving

Range)

CenterlineX=

X

Lower

Plot

UCLmR=

(3.865)(Med

ianMovingRan

ge)

LCLmR=

None

CenterlinemR

=

Median

MovingRange

Step12b

-Constructing

anXmR

Chart

CONTR L C U


61/70

ROLC

HART

VIEWGRAPH1

9

Co

ntrolCh

artZones

1/3distance

from

Centerline

toControl

Limits

LC

L

Centerline

ZONEA

ZONEB

ZONEC

ZONEC

ZONEB

ZONEA

UCL

CONTR

Ru

Cen

UC

LC


62/70

ROLC

HART

VIEWGRAPH2

0

u

le1-InterpretingX-Bar&

RCharts

.

.

.

.

.

.

.

.

.

nterl

ine

CL

CL

ZONEA

OutofLimits

ZONEB

ZONEC

ZONEB

ZONEA

ZONEC

CONTRO

UC

LC

Ce

Ru


63/70

OLC

HART

VIEWGRAPH2

1

.

.

.

.

.

CL

CL

en

terline

ZONEA

ZONEB

ZONEC

ZONEC

ZONEB

ZONEA

2outo

f3

successive

values

inZone

A

ule2-Inte

rpreting

X-Bar&

RCharts

CONTRO

UCL

LCL

Cen

Ru


64/70

OLCHART

VIEWGRAPH2

2

.

.

.

.

LL nte

rline

4outof5

successivevalues

inZone

sA&B

ZONEA

ZONEB

ZONEC

ZONEC

ZONEB

ZONEA

u

le3-Interpreting

X-Bar&

RChart

s

CONTRO

UCL

LCL

Cent

Ru


65/70

OLCHART

VIEWGRAPH2

3

.

.

Lte

rline

8successive

valuesonsam

e

sideofCenterline

u

le4-Interpreting

X-Bar&

RChart

s

ZONEA

ZONEB

ZONEC

ZONEC

ZONEB

ZONEA

CONTRO

X1

X2

X34

R XX-Bar


66/70

OLCHART

VIEWGRAPH2

4

EXERC

ISE1

Values

ofX-BarandRanges

2

4

6

5

6

2

7

5

73

5

6

2

3

4

1

8

3

4

6

6

4

6

6

29

3

2

2

7

4

7

4

6

4

3

4

6

6

1

57

8

5

4

4

2

3

3

5

3

62

5

2

4

7

5

6

4

5

3

6

2

3

6

7

8

9

131415

4

11

12

5

10

12

16

3.5

5.54.5

5

.05.3

4.3

5.34.8

5.0

3.34.53.8

4.84.34.34.0

5

7

4

4

6

3

3

6

4

5

4

3

4

4

4

r

4

CONTRO

AVERAGERANGE

Not


67/70

LCHART

VIEWGRAPH25

ZONEA

ZONEB

ZONEC

ZONEC

ZONEB

ZONEA

EXER

CI

SE1

X-B

ar&

R

Con

trolCh

a

rt

AVERAGERANGE

.

ZONEA

ZONEB

ZONEC

ZONEC

ZONEB

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

7654321963

te:S

olidlinesrepresentthegridusedinthismodule;das

hedlinesseparatezones.

CONTROL

Date

XVal

mR


68/70

LCHART

VIEWGRAPH26

EXERC

ISE2

Value

sofMov

ingRanges

1

2

3

4

5

6

7

8

9

101112131415

16

17181920

8

16

20

21

2

8241916172

419

22261915

2117

221614

ues

13

4

1

204

5

3

1

7

5

3

4

7

4

6

4

5

6

2

CONTROL

INDIVIDUALX

35

30

25

20

15

10

05

15

10

05

0

MOVINGRANGE

Note:S


69/70

LCHART

VIEWGRAPH27

EXER

CI

SE2

X

mRContr

olChart

ZONEB

ZONEA

ZONEC

ZONEC

ZONEA

ZONEB

..

.

.

..

.

.

.

.

.

.

.

.

.

.

.

.

.

..

.

. .

olid

linesrepresentthegridus

edinthismodule;dashed

linesseparatethezonesintheupperplot.

CONTROL

Xm

INDIVIDUALX

35

30

25

20

15

10

05

15

10

05

0

MOVINGRANGE.

OldCo

Note:S


70/70

LCHART

VIEWGRAPH28

EXER

CI

SE2

RControlC

hartRevis

edforInflatedLimits

ZONEB

ZONEA

ZONEC

ZONEA

ZONEC

ZONEB

ontrolLimits

NewControlLimits

Solidlinesrepresentthegridu

sedinthismodule;lightdashedlinesdividethezonesintheupperplot.

OldCenterline

NewCenterline

O

ldControlLimit

NewControlLimit

Documents

Mod10 Control Chart